Categories: Mathematics, Physics.

Given a linear operator \(\hat{L}\) acting on \(x \in [a, b]\), its **fundamental solution** \(G(x, x')\) is defined as the response of \(\hat{L}\) to a Dirac delta function \(\delta(x - x')\) for \(x \in ]a, b[\):

\[\begin{aligned} \boxed{ \hat{L}\{ G(x, x') \} = A \delta(x - x') } \end{aligned}\]

Where \(A\) is a constant, usually \(1\). Fundamental solutions are often called **Green’s functions**, but are distinct from the (somewhat related) Green’s functions in many-body quantum theory.

Note that the definition of \(G(x, x')\) generalizes that of the impulse response. And likewise, due to the superposition principle, once \(G\) is known, \(\hat{L}\)’s response \(u(x)\) to *any* forcing function \(f(x)\) can easily be found as follows:

\[\begin{aligned} \hat{L} \{ u(x) \} = f(x) \quad \implies \quad \boxed{ u(x) = \frac{1}{A} \int_a^b f(x') \: G(x, x') \dd{x'} } \end{aligned}\]

\(\hat{L}\) only acts on \(x\), so \(x' \in ]a, b[\) is simply a parameter, meaning we are free to multiply the definition of \(G\) by the constant \(f(x')\) on both sides, and exploit \(\hat{L}\)’s linearity:

\[\begin{aligned} A f(x') \: \delta(x - x') = f(x') \hat{L}\{ G(x, x') \} = \hat{L}\{ f(x') \: G(x, x') \} \end{aligned}\]

We then integrate both sides over \(x'\) in the interval \([a, b]\), allowing us to consume \(\delta(x \!-\! x')\). Note that \(\int \dd{x'}\) commutes with \(\hat{L}\) acting on \(x\):

\[\begin{aligned} A \int_a^b f(x') \: \delta(x - x') \dd{x'} &= \int_a^b \hat{L}\{ f(x') \: G(x, x') \} \dd{x'} \\ A f(x) &= \hat{L} \int_a^b f(x') \: G(x, x') \dd{x'} \end{aligned}\]

By definition, \(\hat{L}\)’s response \(u(x)\) to \(f(x)\) satisfies \(\hat{L}\{ u(x) \} = f(x)\), recognizable here.

While the impulse response is typically used for initial value problems, the fundamental solution \(G\) is used for boundary value problems. Suppose those boundary conditions are homogeneous, i.e. \(u(x)\) or one of its derivatives is zero at the boundaries. Then:

\[\begin{aligned} 0 &= u(a) = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'} \qquad \implies \quad G(a, x') = 0 \\ 0 &= u_x(a) = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'} \quad \implies \quad G_x(a, x') = 0 \end{aligned}\]

This holds for all \(x'\), and analogously for the other boundary \(x = b\). In other words, the boundary conditions are built into \(G\).

What if the boundary conditions are inhomogeneous? No problem: thanks to the linearity of \(\hat{L}\), those conditions can be given to the homogeneous solution \(u_h(x)\), where \(\hat{L}\{ u_h(x) \} = 0\), such that the inhomogeneous solution \(u_i(x) = u(x) - u_h(x)\) has homogeneous boundaries again, so we can use \(G\) as usual to find \(u_i(x)\), and then just add \(u_h(x)\).

If \(\hat{L}\) is self-adjoint (see e.g. Sturm-Liouville theory), then the fundamental solution \(G(x, x')\) has the following **reciprocity** boundary condition:

\[\begin{aligned} \boxed{ G(x, x') = G^*(x', x) } \end{aligned}\]

Consider two parameters \(x_1'\) and \(x_2'\). The self-adjointness of \(\hat{L}\) means that:

\[\begin{aligned} \int_a^b G^*(x, x_1') \Big( \hat{L} \{ G(x, x_2') \} \Big) \dd{x} &= \int_a^b \Big( \hat{L} \{ G(x, x_1') \} \Big)^* G(x, x_2') \dd{x} \\ \int_a^b G^*(x, x_1') \: \delta(x - x_2') \dd{x} &= \int_a^b \delta^*(x - x_1') \: G(x, x_2') \dd{x} \\ G^*(x_2', x_1') &= G(x_1', x_2') \end{aligned}\]

- O. Bang,
*Applied mathematics for physicists: lecture notes*, 2019, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch".
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